Bochner's theorem (Riemannian geometry)

Template:Short description In mathematics, Salomon Bochner proved in 1946 that any Killing vector field of a compact Riemannian manifold with negative Ricci curvature must be zero. Consequently the isometry group of the manifold must be finite.Template:SfnmTemplate:SfnTemplate:Sfn

Discussion

The theorem is a corollary of Bochner's more fundamental result which says that on any connected Riemannian manifold of negative Ricci curvature, the length of a nonzero Killing vector field cannot have a local maximum. In particular, on a closed Riemannian manifold of negative Ricci curvature, every Killing vector field is identically zero. Since the isometry group of a complete Riemannian manifold is a Lie group whose Lie algebra is naturally identified with the vector space of Killing vector fields, it follows that the isometry group is zero-dimensional.Template:Sfnm Bochner's theorem then follows from the fact that the isometry group of a closed Riemannian manifold is compact.Template:Sfnm

Bochner's result on Killing vector fields is an application of the maximum principle as follows. As an application of the Ricci commutation identities, the formula

$\mathrm{\Delta }X=-\mathrm{\nabla }(\mathrm{div}X)+\mathrm{div}({ℒ}_{X}g)-\mathrm{Ric}(X,\cdot )$

holds for any vector field Template:Mvar on a pseudo-Riemannian manifold.Template:Sfnm As a consequence, there is

$\frac{1}{2}\mathrm{\Delta }⟨X,X⟩=⟨\mathrm{\nabla }X,\mathrm{\nabla }X⟩-{\mathrm{\nabla }}_X\mathrm{div}X+⟨X,\mathrm{div}({ℒ}_{X}g)⟩-\mathrm{Ric}(X,X).$

In the case that Template:Mvar is a Killing vector field, this simplifies toTemplate:Sfnm

$\frac{1}{2}\mathrm{\Delta }⟨X,X⟩=⟨\mathrm{\nabla }X,\mathrm{\nabla }X⟩-\mathrm{Ric}(X,X).$

In the case of a Riemannian metric, the left-hand side is nonpositive at any local maximum of the length of Template:Mvar. However, on a Riemannian metric of negative Ricci curvature, the right-hand side is strictly positive wherever Template:Mvar is nonzero. So if Template:Mvar has a local maximum, then it must be identically zero in a neighborhood. Since Killing vector fields on connected manifolds are uniquely determined from their value and derivative at a single point, it follows that Template:Mvar must be identically zero.Template:Sfnm

References

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